Identity commutator cyclic ladder ($A{B}^{i}y=i{B}^{i-1}y$)

For real-linear operators $[A,B]=I$ and a nonzero vector $y$ with $Ay=0$, induction gives $[A,B^i]=i{B}^{i-1}$, hence the displayed formula. Applying $A^n$ to a nontrivial finite dependence with highest term $c_n{B}^{n}y$ leaves $n!c_{n}y$, a contradiction. Thus $y,By,B^{2}y,\dots$ are linearly independent, and their span is invariant under both operators. Characteristic zero is essential to the factorial argument.